25  Double Integrals in Polar II

25.1 Review

NoteArea of a rose petal

Consider the rose curve \(r = \cos(2\theta)\) as above. What integral do we need to compute the area of one petal?

ImportantBig idea

To evaluate a double integral over a rectangular

TipPolar integration (Stewart et al. 2020, 15.3.2)

If \(f(x,y)\) is a continuous function on a polar rectangle \(\alpha \leq \theta \leq \beta\) and \(g_1(\theta) \leq r \leq g_2(\theta)\), then \[ \iint_R f(x,y) \, dA = \int_a^b \left( \int_{g_1(\theta)}^{g_2(\theta)} f(r \cos \theta, r \sin \theta) \, r \, dr \right) d\theta. \]

Stewart, J., D. K. Clegg, and S. Watson. 2020. Calculus: Early Transcendentals. 9th ed. Cengage Learning.